The pilot reported that while performing touch-and-go maneuvers, he had applied carburetor heat during the second landing. During the takeoff, after the second landing, he turned the carburetor heat off and applied full power. Once airborne, the airplane began to lose power and he manipulated the throttle and carburetor heat levers but was unable to regain power. He chose to make a precautionary landing straight-ahead with partial engine power. During the landing on the remaining runway, the airplane departed the end of the runway, encountered unmaintained snow, and nosed over. The airplane sustained substantial damage to the right wing lift strut and empennage. A post-accident engine examination and test run was accomplished by the NTSB and FAA with no malfunctions or failures revealed. The exact probability of carburetor icing was not able to be calculated for the accident airport due to the closest weather observation station being 8 miles away. The pilot however stated that while landing, there was about 3 inches of fresh snow on the runway, and that he believed the propeller wash likely introduced snow into the intake, which created carburetor ice. He added that he should have left the carburetor heat on longer after the second landing.